Nodal and other properties of the second eigenfunction of the Laplacian in the plane

Author:
Richard L. Liboff

Journal:
Quart. Appl. Math. **63** (2005), 673-679

MSC (2000):
Primary 65Nxx, 35Jxx

Published electronically:
August 3, 2005

MathSciNet review:
2187925

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Abstract | References | Similar Articles | Additional Information

Abstract: Rules are established for the intersection of nodals at a boundary in the plane relevant to the second eigenfunction of the Laplacian. Employing these results together with regularity theorems related to odd reflection of solutions of the Helmholtz equation, as well as a variation of C.S. Lin's analysis, the following theorem is revisited: The nodal curve of the second eigenstate of the Laplacian for bounded convex domains in the plane, with Dirichlet boundary conditions, is a simple curve that intersects the boundary in two distinct points. Application is made to the regular convex polygons with , symmetry and to convex billiards with smooth boundaries.

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Additional Information

**Richard L. Liboff**

Affiliation:
Schools of Electrical Engineering and Applied Physics, and Center for Applied Math, Cornell University, Ithaca, New York 14853-5401

Email:
richaard@ece.cornell.edu

DOI:
https://doi.org/10.1090/S0033-569X-05-00975-3

Received by editor(s):
January 11, 2005

Received by editor(s) in revised form:
February 24, 2005

Published electronically:
August 3, 2005

Additional Notes:
Fruitful discussions on these topics with my colleagues Alfred Schatz, Bradley Minch, Mason Porter and Sidney Leibovich are gratefully acknowledged. I am particularly indebted to Lawrence Payne for sharing his expertise with me in this study.

Article copyright:
© Copyright 2005
Brown University